Aplikace matematiky
Pavel Krejčí
On Ishlinskij's model for non-perfectly elastic bodies
Aplikace matematiky, Vol. 33 (1988), No. 2, 133--144
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33(1988)
APLIKACE MATEMATIKY
No. 2,133—144
ON ISHLINSKirS MODEL
FOR NON-PERFECTLY ELASTIC BODIES
PAVEL KREJCI
(Received December 1, 1986)
Summary. The main goal of the paper is to formulate some new properties of the Ishlinskii
hysteresis operator F, which characterizes e.g. the relation between the deformation and the
stress in a non-perfectly elastic (elastico-plastic) material. We introduce two energy functionals
and derive the energy inequalities. As an example we investigate the equation u" + F(w) = 0
describing the motion of a mass point at the extremity of an elastico-plastic spring.
Keywords: Ishlinskii operator, hysteresis, non-perfect elasticity, energy inequalities, damped
vibrations.
AMS Classification: 73E99, 34K15.
In 1944 Ishlinskii [1] proposed to describe the relation between the internal
stress and the prolongation (Hooke's law) in non-perfectly elastic materials by means
of a hysteresis scheme. A very complete mathematical theory of hysteresis phenomena
has been elaborated by Krasnoselskii and Pokrovskii [2], who introduced the
notion of the Ishlinskii operator and studied its basic properties.
Our aim is to derive further properties of the Ishlinskii operator. In particular,
we introduce potential energies Pt and P2 and prove the corresponding energy
inequalities. Next we investigate the asymptotic behaviour and oscillatory properties
of the solution to the problem (where F is the Ishlinskii operator)
u" + F(u) = 0 , u(0) -= K , u'(0) = 0 ,
describing the free vibrations of a mass point at the extremity of a non-perfectly
elastic spring. We observe a strong dissipation of energy and study the relation
between the rate of decay of the solution and the form of the corresponding hysteresis
loop.
This paper is supposed to be an improved version of the first two parts of [4].
The last part of [4] concerning a nonlinear wave equation will be published later
in another framework.
133
1. HYSTERESIS OPERATORS
We first recall the definition of hysteresis operators (cf. [2], [3]). Let C([0, T])
denote the B-space of all continuous functions v: [0, T] -> R1 with sup-norm
II' l[o,T]» anc * let v e C([0, T]) be piecewise monotone. For h > 0 we define
max {lh(v) (t0), v(t) - h) , te (t0, tt]
- . , v _ / if t? is nondecreasing in [*0, lx] ,
W W " \ i _ _ { Z ^ ) ( O , t < 0 + *}.
te(f0,ti-}
if i> is nonincreasing in \t0, fj] ;
...
(1.1) (i)
, v(0) - h if
00
/*(»)(o)=-(o
\ ( 0 ) + h if
(iii)
»(0) > h ,
if Ko)| = h,
/*00(0««<0-'*(«0(0.
»(0) < - h ;
te[0,T].
The functions /A(v), f,(v) are continuous and piecewise monotone. Moreover, if
v,we C([0, T]) are continuous and piecewise monotone, then
(1.2)
\lh(v) (0 - h(w) (0| =S ||* - w|| [ 0 , r ] ,
*6 [0, T] .
This property enables us to define the values of lh(v) (fh(v)) for arbitrary v e C([0, T])
as lim lh(vn) (fh(vn)), where vn e C([0, T]) are piecewise monotone and \\vn — ^Uro.T] "*
n-*oo
-> 0. Thus /,,,f. become Lipschitz continuous operators from C([0, T]) into C([0, T]).
Further, let x e 1/(0, co) be a function such that
(1.3)
(i)
*(x)_0
(ii)
Va > 0 ,
a.e.,
x(x) dx > 0 .
We denote
(1.4)
<?(*) = f'f"x(Z) dt dx , #(z) = [cp(x) dx .
J 0J x
Jo
The Ishlinskii operator is given by the formula
*00
(1.5)
F(v) (t) =
fh(v) (t) X(h) dh for
v e C([0, T]),
f e [0, T] .
Jo
Properties of hysteresis operators
(1.6) (i) lh,fh, F are continuous and odd operators C([0, T]) -> C([0, T]),
(ii) for
v, w e C([0, T]) we have
\F(v)(t) - F(w)(0| _ 2p(||t, 134
W||r(M])
_ 2p'(0+) |* - Hk*]>
(iii) for v absolutely continuous, F(v) is absolutely continuous; if v'(t) = 0,
then (F(v))' (t) = 0 and if both v'(t) * 0, (F(v))' (t) exist, then F(v)' (t) . v'(t) > 0.
Moreover, the inequality
<p'(\\v\\lo,n).\v'(t)\<\(F(v))'(t)\<2(p'(0+)\v'(t)\
holds almost everywhere.
(iv) Let v e C([0, Tj) and v(tm) = - |i>||[0>r] (or v(tM) = ||t>||[0>r]).Then/„(t>) (tm) =
= -\\fh(v)lio,T) = - min{/t, -t>(fm)} (or fh(v)(tM) = |]/*(»)||[0>T] = mm{h,v(tM)},
respectively) for all h > 0, and F(v)(t,„) = -<p(-v(tm))(F(v)(tM) = <p(v(tM))).
In particular, ||F(t>) (t)||[0>r] = HHIr.o.r])- Further, let tm < T and let v be
nondecreasing in \tm, t t ] (or nonincreasing in in \tM, f.]). Then fh(v) (t) =
= fh(v) (Q + min {2h, v(t) - v(tm)}, h > 0, t e \tm, tt] (fh(v) (i) = fh(v) (tM) — min {2/j, v(tM) — v(t)}, h > 0, f e \tM, r x ], respectively).
(v) Let 0 = f0 < J, = F, t>.(0 ^ t>.(t) ^ t^f.) (or t>.(*0) ^ t>,(t) = ».(*.)) for
f 6 [t 0 ,*.], / = 1, 2, 0l (r t ) = t>2(ft), fe = 0, 1,/»(».) ( 0 = A(t>2) (t0). Then/,^,) (f.) =
= fh(v2)(h).
(vi) Let 0 ^ f0 < tt < t2 = T, v nonincreasing in \tt, f 2 ], t>(t2) ^ »(f0),
/*(«)(ti) =-/*(») (to) + min {2h,v(tt) - v(t0)}. Then for ( e [ ( „ r2 ] we have
/*(») (t) = fh(v) (ti) _ min {2h, v(tt) — v(t)} (analogously, if v is nondecreasing
in [f., r 2 ], t>(f2) < v(t0),fh(v) (t,) = fh(v) (t0) - min {2h, v(t0) - vfa)},
then
f„(v) (t) = fh(v) (ti) + min {2h, v(t) - v(tt)} for t e [, lt f2]).
(vii) Let fh(v) (t) = fh(v) (t0) ± min (2/t, \v(t) - v(t0)\} for t e [f,,t 2 ] and for all
h > 0. Then F(v) (t) = F(v) (t0) ± 2<p(\\v(t) - v(t0)\), te\tu t2] .
(viii) Let it, t> be absolutely continuous in [0, T~\. Then fh(u),fh(v) are absolutely
continuous and ([(/*(«))' (t) - u'(t)] - \(fh(v))' (t) - v'(tj]) (fh(u) (t) - fh(v) (t)) < 0
is satisfied for almost every h > 0 and almost every t e (0, T). In particular,
J V ( « ) (o-) - F(t>) (<r)) (u'(<r) - „'(<-)) d<r
=
(t) - fh(v) (t))2 - (/*(«) (s) - fh(v) (s))2] # ) dh
= i j\fh(u)
holds for every s, t,0 = s < t = T.
(ix) Let u, v be absolutely continuous and co-periodic functions. Then fh(u), fh(v),
F(u), F(v) are co-periodic for t = co. Let us assume xQ1) > 0 a-e- i n (0> G0) a n d
i:
(E(«)(0-E(»)(0)(«'(t)-t>'(t))dt = o.
Then u' = v' almost everywhere.
Proof of (1.6). Parts (i)-(vii) in this form are proved in [3]. The assertion of
(viii) corresponds to Sec. 36.3 of [2]. The co-periodicity of fh(u), F(u) for an co-periodic
function u is also proved in [2].
135
Let us assume that the relation in (ix) is satisfied, i.e.
rcxDt*2co
(/*(«) (t) - /*(») (t)) («'(t) - v'(t)) dt X(h) Ah = 0 .
J 0 Jco
By (viii) we have for almost all h > 0 and a.e. t > 0
(/»(«) (t) - /.w (t)) MO - "'(o = i ~ [/*(«) (o - m (t)]2.
dt
We conclude that
(1.7)
([(/„(«))' (0 - «'(*)] - [(/,(,))' (0 - v'(t)]) (f„(u) (t) - fh(v) (0) = o
is fulfilled for a.e. h > 0 and t ^ co.
Let us choose f ^ 2co such that M'(t), vf(t) exist. Let e.g. w'(t) > 0 and put t =
= max { r e [0, of],
|M(T)| = ||tt|| [ 0 > w l }.
For
w(t) > 0
we
put
tt = t,
t2 =
= max {T G [l l 5 tx + of], M(T) = min {w(?/), rj e [tl91± + of]}}. For w(t) < 0 we put
t0 = t, t± = max {T G [t 0 , t0 + of], M(T) = max {u(rj), rj e [t0, t0 + of]}}, t2 = t0 +
+ OJ. By induction we define
(1.8)
t2J+i = max {T G [t2J, t], M(T) = max {u(rj), rj e [t2J, t]}} ,
t2J+2 = max {T G [t2J+lit],
u(r) = min {u(n), rj e [t2J+u
t]}} ,
until tk = t.
The assumption M'(t) > 0 ensures that there exists n ^ 1 such that t = l2n+1*
Following (1.6) (iv)-(vi) we have for all h > 0 andj = 1 , . . . , n
(1-9)
fh(u) (t2J+i)
= A(M) (t 2 ; ) + min {2/i, tt(r2y+1) - M(t2;)} ,
fh(u) (t2J) = fh(u) (t2J-f)
- min {2/z, u(t2J-f)
- u(t2$
.
The following lemma is obvious.
(1.10) Lemma. Let u: [a, b] -» R1 be absolutely continuous, u(a) :£ M(T) ^ M(b)
(u(a) ^ M(T) ^ u(b)) for T G [a, b]. Put M(T) = max {w^), a ^ rj S ?} (tt(?) =
= min {u(rj), a % rj :§ T}, respectively). Then u is absolutely continuous and if we
denote M = . { T G(<3, b), M'(T) + 0], then M(T) = M(T) for every t e M and M'(T) =
= M'(T) for a.e. T G M.
(1.11) Lemma. Let tt be absolutely continuous, let u'(t) > 0, (F(u))'(t) exist, and let
{tk, k = 1, ..., 2n} be the sequence (1.8). If 2h > M(^) — u(t2n), then (fh(u))' (t) =
= ur(t), if 2h < u(t) - tt(l2n), then (fh(u))' (t) = 0.
P r o o f of (1.11). The first part is obvious. For 2h < u(t) — u(t2n) we introduce
the function u from (1.10) in [t2n, t]. For every T < t we have M(T) < u(t) = M(T) +
+ §1 ur(rj) drj, hence meas ([T, t] n M) > 0. Thus we can find T 0 < t such that
M(T0) = M(T0) and for T G (T 0 , l) we have M(T) — w(t2,,) > 2h, M(t) — M(T) < 2h.
For T > T0 we have fh(u) (T) = ft and fh(u) (T) — f,,(w) (T) = M(T) — M(T). Similarly
136
we find zx > t such that u^) = u(xx) < u(t2n-^) and W (TJ) - u(x) < 2h for
xe [t9 xt). The assertion of (1.11) follows from the fact that u'(t) = u'(t).
•
The function v in (1.7) is nonconstant. Indeed, if v(t) === Vo g 0, then for sufficiently small h > 0 we have fh(u) (t) = h, (fh(u))' (t) = 0, fh(v) (t) g 0 and this
contradicts (1.7). For v0 > 0 we find t' such that u'(t') < 0 and we proceed as above.
Let us denote by {sk) the sequence (1.8) corresponding to v. If {sk} is infinite,
we have for sufficiently small h > 0:fh(u) (t) = h, (fh(u))' (t) = 0, and for sufficiently
large j : - h < fh(v) (s2j) < fh(v) (t) < fh(v) (s2j+ -) < h, v'(t) = (fh(v))' (t) = 0,
which contradicts (1.7).
Let us assume t = s2m for some m > 1. Then v'(t) ^ 0 and for sufficiently small
h > 0 we again obtain a contradiction to (1.7).
We conclude that t = s2m+1 for some m ^ 1 and relations analogous to (1.9)
hold for v and {sk}. Moreover, for the same reasons as above the assumption v(t) —
— v(s2m) < 2h < u(t) ~ W(^2M) leads to a contradiction.
Thus we have proved (notice that the case u'(t) < 0 is analogous: the mapping
f7jis odd):
(1.12)
Lemma. Let the assumptions
of (1.6) (ix) be satisfied and let u'(t), v'(t),
(F(u))' (t), (F(v))' (t) exist, t = 2co.
If u'(t) > 0, then t = t2n+1 = s2m+1 and
(i) u(t) - u(t2n) < v(t) - v(s2m) ,
if u'(t) < 0, then t = t2n = s2m and
0 0 u(t2n-i) ~ u(t) £ v(s2m-i)
Let us prove
(1.13)
=
S
2m,
- v(t).
Lemma. Let the assumptions
U
(t) ~ u(t2n)
=
V
(t) ~
of (1.12) be satisfied, u'(t) > 0. Then t2n =
V S
i 2m)-
P r o o f of (1.13). A. Assume s2m < t2n^t. Following (1.10) we construct the function u in [*2„-1? Lz,.]. Put r0 = min{xe[t2n„l9t2n],
u(x) = u(t2n)}. For every
T < r 0 we have meas [T, r 0 ] n M > 0, hence there exists a sequence Xj S r0, u'(x3) =
= u'(xj) < 0, v'(xj) exist. Following (1.10), (1.12) (we have Xj > tl9 which is sufficient
for (1.12)) we find aje(s2m, Xj) such that u(t2n_1) — u(xj) S v(aj) — v(xj). We can
assume aj -> a0 and passing to the limit as j -> oo we obtain
(1.14)
"('2«-i) -
u
(hn) ^ v(°o) ~ v(r0) ,
where v(t) ^ v(a0) ^ v(cx) ^ v(r0) for a e [a0, r 0 ] . Next we put
rt = max {a e [a0, t]; v(a) = min {v(rj)9 rj e [a0, t]}} ,
r 2 - min {a e [rl9 t], v(a) = v(a0)} .
We have r2> r1>z r0, v(rx) ^ v(r0). Using the function v in [r l 5 r 2 ] we find sequences a} /" r 2 , xj e (t2n-l9 a3) such that V'(GJ) > 0, u'(Gj) exist and
137
V(GJ) - v(rt) ^ U(GJ) - u(xj) S u(an) - u(*2n) ,
v(r2) - v(rx) g u(r2) - u(l2n) .
From (1.14) we conclude u(r2) ^ u(t2n-i),
which contradicts (1.8).
B. Assume t2n_1 < s2m < t2n. Following (1A0) we construct v in [s 2 m , t]. Obviously v(t2n) > v(s2m). Put r 0 = min {G e [s 2 m , t2n~], V(G) = v(t2n)}. We find again
the sequences Gj J* r 0 , xje(t2n_1,GJ),
Ty -> T 0 such that v'(0"j) > 0, U'(GJ) exist
and V(GJ) - v(s2m) ^ u(r/y) - u(Ty),
(1.15)
v(r0) - v(s2m) g u(r0) - u(T0) .
We see that r 0 < £2n, u(f2n) ^ u(T0) :§ W(T) ^ u(r0) for r e [T 0 , r 0 ] . Put r x =
= max {T G [T 0 , r 2 J ,
u(x) = max {u(rj), n e [T 0 , *2J}}>
r 2 = min {T e [rl9 t2n~],
u(x) = u(T0)}. We define the function u in [r 1? r 2 ] and find the sequences Ty /* r 2 ,
Gj e (r 0 , Ty) such that u'(xj) < 0, v'(xj) exist and
u(rx) - u(tj) ^ v(Gj) - v(Ty) ^ v(r0) - v(xj) ,
u(rt) - u(r2) g v(r0) - v(r2) .
Now (1.15) implies v(r2) ^ v(s2m), which contradicts (1.8).
C. Assume s2m ^ l2n. We define v in [s 2 m , t] and similarly as above we obtain
v(t) - v(s2m) g u(t) - u(t2n) .
For s 2m > l2n we use A, B, where u is replaced by v and vice versa. Lemma (1.13)
is proved.
•
We can complete the proof of (1.6) (ix). For a fixed t ^ 2co such that u'(t), v'(t)
exist, u'(t) > 0 there are two possibilities:
(a) Vs e (t2n, t), u(s) < u(t). In [t2n, t] we define u and find Ty S1 t as above.
Following (1.13) we have u(x3) — u(t2n) = v(Ty) — v(t2n), hence
u(l) - u(xj)
f -
T,
=
v(Q - v(Ty)
l - Ty
which implies v'(t) = u'(t).
(b) 3s e (l 2n , l), u(s) = u(r). Then the derivative (F(u))' (t) does not exist. Since
the set of such points is of measure zero, the proof of (1.6) (ix) is complete.
R e m a r k . In general, in the nonperiodic case the assumption that (1.7) holds
for alJ h > 0 and t > 0 does not imply u'(t) = v'(l): it suffices to choose u bounded
and non-decreasing, v constant, v(t) = v0 > ||u||.
138
Potential energies
Let us denote by Wk'p(0, T)9 k = 1, 2 , . . . , 1 g p g oo the usual Sobolev space.
We define the expressions
(i) P 1 (w)(0 = i j / , 2 ( « ) ( 0 ^ ) d h
(1.16)
(ii)
(1.17)
P 2 (w)(0 = K ^ ) y ( 0 " ' W
^r
for
ueC([0,T]),
u e ^ ^ O J ) .
Energy inequalities.
(i) Let u e Wx^(0, T). Then P±(u) e WU1(0, T) and
(Pi(u))' (0 -
F u
( ) (0 u'(t) S 0
a.e. in
[0, T] .
(ii) Let us assume that there exists a positive decreasing function y such that
for all K > 0 and for almost all x e (0, K] we have x(x) ^ ?(-^)- Then for every
ue W2'1^, T) the inequality
P2(u)
(s) -
P2(u)
=
holds for a.e. t,s,0
(t) -
!\F(U))'
(O) U"(O) do <;
-h(\\u\\i0jjy(cr)\3do
^ t < s ^ T.
Proof. Part (i) follows immediately from (1.6) (viii).
(ii) Let 0 ^ t < s S T be given. The set {a e (t, s), u'(o) + 0} is a countable
union of open intervals. Hence it suffices to assume that u is strictly monotone in
[t, s]. Let u be increasing in [t, s] (the other case is symmetric). Put l = max {T G [0, t],
\U(T)\ = ||w|| [0>f] }, and assume
A. t < t. We put t = t0 for u(t) < 0 and t = tx for u(t) > 0. Let {tk} be the
sequence (1.8), k = 0, 1, . . . , F o r j such that u(t2J+1) e[u(t),u(s)]
we denote by
T
2j+i G [t, s] the point where U(T2J+1) = u(^2j-+1). In the case t = l0, u(s) > \u(t0)\
we find T 0 e (t, s), U(T0) = — u(t0). The singular points T2J+1 are those where (F(u))'
and consequently P2(u) are not defined. Thus we deal only with the case t, s + T2J+1.
Moreover, we can assume that t, s are separated by at most one point T 2 J - + 1 . In the
opposite case we replace the interval [t, s] by a countable union of intervals having
the required property (notice that the only possible limit point of T2J+1 is t, and in
that case we have u (t) = 0, |F(u))' (o)\ ^ c|u'((r)| -» 0 for a \ t, a + T2J+1). By
(1.6) (iv) —(vii) the following cases are possible:
(1.18)
(i) F(u) (o) = F(u) (t2n) + 2<p(i(u(a) - u(t2n))),
(ii)
F(u) (a) = F(u) (t0) + 2(p(i(u(o) - u(t0))) ,
(p(u(o)), a e [T, S] ;
ae[t9s];
a e [t, T) ,
139
(iii)
F(u) (o) = F(u) (t2n+2) + 2(p(i(u(o) - u(t2n+2))) , o e [t, T) ,
F(u) (t2n) + 2cp{i(u(o) - u(t2„))),
o e [T, S] ,
where u(t2„) < u(*2„ + 2 ) .
B. ! = L For u(t) < 0 we have (1.18) (i) or (ii), for u(t) > 0 we obtain
(1.18)
(iv) F(u)(o)
= <p(u(o)),
os[t,s].
The proof follows from an easy integration by parts.
(1.19) R e m a r k . Integrating directly in (1.16) (i) we obtain another expression for
Pi(u):
Pi(u) (t) = u(t) <p(u(t)) - $(u(t)) if F(u) (t) = <p(u(t)) ,
Px(u) (t) = P,(u) (to) + i(u(t) - u(t0)) (F(u) (to) + F(u) (t))
if
F(u) (t) = F(u) (t0) + 2(p(i(u(t) -
u(l0))),
and similarly for u nonincreasing.
2. AN ORDINARY DIFFERENTIAL EQUATION
Let US consider the problem
(2.1)
(i)
u" + F(u) = 0 ,
(ii)
u(0) = K > 0 , u'(0) = 0 ,
where F is given by (1.3), (1.5).
(2.2) Theorem. There exists a unique classical solution u: [0, oo) -> R1 to the
problem (2.1) (i), (ii) and this solution has the following properties:
(i) There exists a sequence 0 = t0 < tt < t2 < ... such that (—1)" u is decreasing
in[tn9 tn+1~] and
lim(tn+l
-tn)=
*
V ^ (
n-^oo
(ii)
There exist positive decreasing functions
0 +
.
)
R(t), g(t) such that
KmR(t) = 0,
lim^iM)
f-»oo
f~>oo
=
0
t
and
e(t)^\u'(t)\
+
\F(u)(t)\SR(t).
Proof. The existence and uniqueness of the solution follows from the Lipschitz
continuity (1.6) (ii) of F. The total energy of the system is given by
(2.3)
140
E1(l) = iu'2(t)
+
P1(u)(t).
From (1.17) (i) we immediately have E[(t) __\ 0 a.e. Let us denote F0 = F(w) (0) =
= (p(K). There exists t_ > 0 such that u is decreasing in [0, t_~\. In the case u(t_) <
< — K we have
£i(ti) = Px(u) (h) > Pt(u) (0) = £,(0) ,
which contradicts the energy inequality. Hence u(t_) __ —K, u'(t_) = 0 and (cf.
(1.6) (iv)) F(u)(t) = F0 - 2<p(i(K - u(t))) for le[0, t_]. Consequently, ut2(t) =
= 2F0(K - u(t)) - 8<l>(i(K - u(t))). Putting z(t) = i(K ~ u(t)) we obtain z'(t) =
= V(F0z - 2 ^(z)), z(0) = 0.
Let 0 < z_ < K be the unique positive root of the equation F0z = 2 $(z). We
have z(li) = z_ and ^'(zt) > 0. Put Kt = K - 2z! and Fx = -F(w)(^) =
= -F(u)(t_) - -F0 + 2<p(z_) = -(2$(z_)lz_) + 2<p(z_) > 0.
We continue by induction, finding the sequences z„, Fn, tn, Kn such that 0 = t0 <
< * - . < . . . , 11(0 = K„, ti'fo) = 0, Fw = (-1)" F(w) ( 0 > 0, z„ +1 = i ( - l ) " .
. (K„ - K „ + 1 ) , Fnzw + 1 = 2 # ( z „ + 1 ) , and for te[frt, ^ + 1 ] the function (~l)n u is
decreasing and F(u) (t) = (~l)n [Fn - 2 ^ ( - l ) " ^ - ii(f)))]. In particular,
^*+i = --?,, + 2(p(zw + 1 ), hence
Fn+1 = 2 ( 2 ^ - ± l } _ ^ ( ^ = _A_ r + 1 p , z ( i , ) d i , d « > 0.
z
\ Z»+l
/
n+lJo Jo
Let us introduce the function a: [0, 2<p(+oo)) -> K1, a(x) = 2(x — (p(z)), where
x = 2 $(z)/z. We have a(0) = 0,
Fu-
'w - (2 - ^ y - o**»«) (I? *® df)_1 > °
and lim a'(x) = 0. Moreover, (x(x)jx < 1 for every x > 0. Put
JC-*0 +
r( \
r° ^
J* «(£)
We see that G is increasing, lim G(x) = — oo and lim log xJG(x) = lim a(x)jx = 0.
x
We have
~*0+
*"*0+
*-° +
«(Fn)
F Fn+1
G(Fn) - G(Fn+1) = - "
<Fn+1 + dn(Fn - Fn+1))
a(Fn+1 + 0n(Fn - Fn+.))
for some 6n e [0, 1], hence
i<r(F\
r(p
\ < -iEA
«(FJ
1 < G(F„) - G(Fn+1) <
- —
—— .
a(E n + 1 )
a(E„ - a(E„))
We immediately obtain G(Fn) ^ — n, hence F„ ^ G _ 1 (—n) -* 0 (and therefore
z„ -* 0) as n -» +oo. Further,
a(x - a(x)) _
lim-i Ґ «'(č)dČ =
a(x)
*-0+ «WJ»-ф)
141
and consequently
Fn = G - 1 (-a 0 /i)
(2.4)
for some a0 > 1 .
Let us estimate the difference t„+1 — tn. We have
t
n+±
+i
=
r
t
n
Jo
dz
— -Zn+i r
J(Fnz~2<P(z))
da
V ^ J o V M v ^ ^ ^
On the other hand, the relation
a <P(z) - <P(az) = a f
|
q>'(tj) dt] d£
Jo Jat
implies
i <p'(z) z2a(l ~a)^a
*(z) - * ( « ) ^ \ <p'(0+) z2a(l - a) .
This yields
(2-5)
- T T ^ T T ^ < » - - <» =
=
V(y(o+)r
" V(^n+1))'
and (2.2) (i) is proved.
Let us now choose t e (t„, tn+t), n = 1. We have w'2(t) = 4(F„ z(t) - 2 #(z(f))) and
W ( f =
ft^^(f,
where z(t) = i(-l)» (K„ - u(t)). We obtain
|w'(r)| + \F(u)(t)\ = ClFn S c2Fn+1 (notice that
\[mlz±l
n +
" °° *»
=l
and t^ tn+l ^ n(n + *) ) ,
vV(zi)
hence
KOI + |Jt«) (01 = c2 G - » ( - vMfi) A
On the other hand,
2 z(ř)
= Ғ 2 - 2(2 - e) Ғ„ ę(z) + 4 <p2(z) - 4єę(z) Ф(z)
Substituting <P(z) ^ xz <p(z) for z e [0, z j , where % = (1 - <p'(z^)j2 <p'(0+)) < 1,
we obtain
(F(u) (t))2 + is <p'(0+) (u'(t))2 ^ F2 - 2(2 - e) Fn <p(z) + 4(1 - *,) <p2(z) ^
^ c3FB2
provided e < 4(1 - %). Therefore (cf. (2.4), (2.5)).
\F(u) (0| + K(0| ^ c4G-> ( - W ^ P + ) \
142
Weput
,(o = oG-i l- ^M^±),), m = ^ (- 4&à <)
Indeed,
r-+oo
t
a0 X /V(0+) *->o + G(x)
and the proof is complete.
(2.6) Examples, (i) Let us assume that for every K > 0 and for almost all x e (0, K]
we have
r(K) xq = x(x) = y(K) xp
for some p = q > — 1, where y = 0 does not vanish identically. We obtain
Clr^
+q
=
\uf(t)\ + |F(tt)(r)| _ c2r^+*>
(cf. [4] for p = q = 0).
(ii) Theorem (2.2) does not imply u(t) -> 0 for t -> + oo. Let, for example, $(x) =
= cx 1+a for x = x0, a e (0, 1). For sufficiently large K we have F0 = c(l + a) Ka,
zx = (a + l/2) 1/a K. By induction we obtain z w+1 = a1/az„ until zn+l < x0. We have
2N
U^ = limw(t) = K + 2 £ (—l)nzw. The sequence {zn} is decreasing, hence the
t-Kx>
n=l
inequalities
2N
0 ^ K + 2X(-l)"z„-U o o
=
2z2N
/I-=l
hold for all N > 0.
Put
^i-(«±iY*___>0.
V 2 J
1 + a1/x
For e e (0, 1) we find N > 0 such that
BT(TS-)°—
Choose K such that z2N = x0. Then we have A(l + s) > (UjK) > A(l - e).
Therefore lim (U^JK) = A. Notice that in general cases the problem of determinaK-*oo
tion of Uoo remains open,
(iii) In the last example the energy (2.3) of the solution of (2.1) does not tend
to zero as t -+ +00. By (1.19) we have E^ = lim Ex(t) = Kcp(K) - <P(K) (-+00
00
~" T,zn+i(F„ — Fn+1). A computation analogous to (ii) yields
n= 0
lim E„
K-*oo cKi+«
V 2 1
1 - cc1 + 1/ «
143
1 + 1/a
(notice that for f(x) - x
we háve [f(x) - /(y)]/(* - y) > / ' [ ( * + y)/2]
provided x 4= y). This shows that the initial mechanical energy is not completely
dissipated. The quantity E„ corresponds to the inaccessible remainder of the potential
energy.
References
[1] A.IO.
Hui/IUHCKUŮ: HeKOTOpBIe npHMeHCHHÍI CTaTHCTHKH K OHHCaHHK) 3aKOHOB jj;ec[)opMHpOBa-
HHfl Teji, H3B. AH CCCP, OTH, 1944, Ns 9, 583-590.
[2] M. A. KpacHOce/ibCKuů, A. B. IIoKpoecKuů: CHCTCMBI C rncTepe3HCOM. MocKBa, HayKa, 1983.
[3] P. Krejčí: Hysteresis and periodic solutions of semilinear and quasilinear wave equations.
Math. Z. 193 (1986), 247-264.
[4] P. Krej"čí: Existence and large time behaviour of solutions to equations with hysteresis.
Matematický ústav ČSAV, Praha, Preprint no. 21, 1986.
Souhrn
O IŠLINSKÉHO MODELU PRO NE ZCELA PRUŽNÁ TĚLESA
PAVEL KREJČÍ
Hlavním obsahem práce je formulace některých nových vlastností hysterezního operátoru
Išlinského, který vyjadřuje např. vztah mezi deformací a napětím v ne zcela pružném (pružně
plastickém) materiálu. Jsou zde definovány dva funkcionály energie a odvozeny energetické
nerovnosti. Jako příklad j e zkoumána rovnice u" + F(w) = 0, která popisuje volné kmity hmot­
ného bodu na pružně plastické pružině.
Pe3K>Me
o MOAEJIH HnuiHHCKoro fljiii HE BnojiHE ynpyrHX TEJI
PAVEL KREJČÍ
rjiaBHOH nejibio pa6oTBi HBJuieTCfl ýopMyjiHpoBKa HeKOTOptix HOBBIX CBOEÍCTB rHCTepe3HCHoro
onepaTOpa HnuiHHCKoro F, KOTOPBIH BBipaacaeT Hanp. cooTHomenHe Meayry flecfropMaimeň H
HanpJDKemieM B He Bnojme ynpyroM (ynpyro-njiacTHHecKOM) MaTepnajie. BBOAJITCH JXBSL ^ymcuHOHana 3HeprHH H AOKa3BiBaK>TCfl 3HepreTHHecKHe HepaBeHCTBa. B KanecTBe npHMepa nccjie^yeTCH
ypaBHeHHe ď + F(u) — 0, onncbiBaronieeflBH5KeHHeMaccoBOH TO^KH Ha ynpyro-nnacra-ieCKOH
npyacHHe.
Authoťs
Praha 1.
144
address: RNDr. Pavel Krejčí, CSc, Matematický ústav ČSAV, Žitná 25, 115 67